In Friedman's urn model ($a_{0} = 1, b_{0} = 0, a = d = 0, b = c = 1$), we have for all integers $n$ and all $t \in ]0,1[$, $$g_{n}(t) = \frac{1}{n!} \sum_{p=1}^{+\infty} p^{n} t^{p} (1-t)^{n+1}.$$ Fix an integer $n \geqslant 2$.
Using the result of question 30 and by expanding $(1-t)^{n+1}$, determine the Taylor expansion of $g_{n}$ to order $n$ at 0.

Let $P ( X )$ be a monic polynomial of degree $d \geqslant 1$ with complex coefficients which we write in the form: $$P ( X ) = a _ { 0 } + a _ { 1 } X + a _ { 2 } X ^ { 2 } + \cdots + a _ { d - 1 } X ^ { d - 1 } + X ^ { d }$$ We assume that $a _ { 0 } \neq 0$. We denote by $\lambda _ { 1 } , \ldots , \lambda _ { d } \in \mathbb { C }$ the roots of $P ( X )$ (with multiplicity). For all integers $n \geqslant 1$, we define: $$N _ { n } = \lambda _ { 1 } ^ { n } + \lambda _ { 2 } ^ { n } + \cdots + \lambda _ { d } ^ { n }$$
We define the function $f : \mathbb { R } \backslash \left( \mathbb { R } \cap \left\{ \frac { 1 } { \lambda _ { 1 } } , \ldots , \frac { 1 } { \lambda _ { d } } \right\} \right) \rightarrow \mathbb { C }$ by $f ( x ) = \frac { Q ^ { \prime } ( x ) } { Q ( x ) }$.
Show that there exists $r > 0$ such that $f$ is expandable as a power series on $] - r , r [$, and that the power series expansion of $f$ at 0 is written: $$\forall x \in ] - r , r \left[ , \quad f ( x ) = - \sum _ { n = 0 } ^ { \infty } N _ { n + 1 } x ^ { n } \right.$$

Let $P ( X )$ be a monic polynomial of degree $d \geqslant 1$ with complex coefficients which we write in the form: $$P ( X ) = a _ { 0 } + a _ { 1 } X + a _ { 2 } X ^ { 2 } + \cdots + a _ { d - 1 } X ^ { d - 1 } + X ^ { d }$$ We assume that $a _ { 0 } \neq 0$. We denote by $\lambda _ { 1 } , \ldots , \lambda _ { d } \in \mathbb { C }$ the roots of $P ( X )$ (with multiplicity). For all integers $n \geqslant 1$, we define: $$N _ { n } = \lambda _ { 1 } ^ { n } + \lambda _ { 2 } ^ { n } + \cdots + \lambda _ { d } ^ { n }$$ Let $Q(X)$ be the reciprocal polynomial of $P(X)$ defined by $Q(X) = X^d P\left(\frac{1}{X}\right)$.
We define the function $f : \mathbb { R } \backslash \left( \mathbb { R } \cap \left\{ \frac { 1 } { \lambda _ { 1 } } , \ldots , \frac { 1 } { \lambda _ { d } } \right\} \right) \rightarrow \mathbb { C }$ by $f ( x ) = \frac { Q ^ { \prime } ( x ) } { Q ( x ) }$.
Show that there exists $r > 0$ such that $f$ is expandable as a power series on $] - r , r [$, and that the power series expansion of $f$ at 0 is written as: $$\forall x \in ] - r , r \left[ , \quad f ( x ) = - \sum _ { n = 0 } ^ { \infty } N _ { n + 1 } x ^ { n } \right.$$

We define a sequence $(a_n)_{n \geqslant 1}$ by setting $$\forall n \in \mathbb{N}^*, \quad a_n = \frac{(-n)^{n-1}}{n!}.$$ We define, when possible, $S(x) = \sum_{n=1}^{+\infty} a_n x^n$, with radius of convergence $R$. It has been shown that $S(x) = W(x)$ for all $x \in ]-R,R[$. Does this result remain true on $[-R, R]$?

Let $f \in L^1(\mathbb{R})$, $\lambda \in \mathbb{R}_+^*$ and let $g$ be the function from $\mathbb{R}$ to $\mathbb{C}$ such that $g(x) = f(\lambda x)$ for all real $x$. Show that $g \in L^1(\mathbb{R})$ and, for all real $\xi$, express $\hat{g}(\xi)$ in terms of $\hat{f}$, $\xi$ and $\lambda$.

Given three real numbers $a, b$ and $c$ with $c \in D$, the Gauss hypergeometric function is defined by $$F_{a,b,c}(x) = \sum_{n=0}^{+\infty} \frac{[a]_n [b]_n}{[c]_n} \frac{x^n}{n!}$$ Show that the power series $\sum \frac{[a]_n [b]_n}{[c]_n} \frac{x^n}{n!}$ is hypergeometric and specify associated polynomials.

Given three real numbers $a, b$ and $c$ with $c \in D$, the Gauss hypergeometric function is defined by $$F_{a,b,c}(x) = \sum_{n=0}^{+\infty} \frac{[a]_n [b]_n}{[c]_n} \frac{x^n}{n!}$$ Conversely, prove that the set of hypergeometric series associated with the polynomials obtained in the previous question is a vector space for which we will give a basis and specify the dimension.

Given three real numbers $a, b$ and $c$ with $c \in D$, the Gauss hypergeometric function is defined by $$F_{a,b,c}(x) = \sum_{n=0}^{+\infty} \frac{[a]_n [b]_n}{[c]_n} \frac{x^n}{n!}$$ Determine the radius of convergence of the power series $\sum \frac{[a]_n [b]_n}{[c]_n} \frac{x^n}{n!}$.

For every integer $n \geqslant 1$, we denote $C_{n}$ the number of well-parenthesized words of length $2n$. We set by convention $C_{0} = 1$.
Show that, for every natural integer $n$, $C_{n} \leqslant 2^{2n}$. What can we deduce for the radius of convergence of the power series $\sum C_{k} x^{k}$?

For every integer $n \geqslant 1$, we denote $C_{n}$ the number of well-parenthesized words of length $2n$. We set by convenience $C_{0} = 1$.
Show that, for every natural integer $n$, $C_{n} \leqslant 2^{2n}$. What can we deduce for the radius of convergence of the power series $\sum C_{k} x^{k}$?

We consider a power series $\sum_{n \geqslant 0} \alpha_n z^n$, with radius of convergence $R \neq 0$ and with $\alpha_0 = 1$. We denote by $S$ the sum of this power series on its disk of convergence.
Show that there exists a real number $q > 0$ such that $\forall n \in \mathbb{N}, |\alpha_n| \leqslant q^n$.

Using the results of questions 28--30, show that there exists a unique complex sequence $(b_n)_{n \in \mathbb{N}}$ and a real number $r > 0$ such that, for all $z \in \mathbb{C}$, $$0 < |z| < r \Rightarrow \frac{z}{\mathrm{e}^z - 1} = \sum_{n=0}^{+\infty} \frac{b_n}{n!} z^n.$$

Let $z \in D$. We agree that $p _ { n , 0 } = 0$ for all $n \in \mathbf { N }$. By examining the summability of the family $\left( \left( p _ { n , N + 1 } - p _ { n , N } \right) z ^ { n } \right) _ { ( n , N ) \in \mathbf { N } ^ { 2 } }$, prove that
$$P ( z ) = \sum _ { n = 0 } ^ { + \infty } p _ { n } z ^ { n }$$
Deduce the radius of convergence of the power series $\sum _ { n } p _ { n } x ^ { n }$.

We assume that $|\lambda| \notin \{0,1\}$ and that $\rho(f) > 0$. Using the result of question (22), conclude that the power series $h$ and $H$ of part E have strictly positive radius of convergence.

For $p \notin \mathbb{N}^*$, $p \neq 0$, let $f(x) = \sum_{n=0}^{+\infty} a_n x^n$ be a non-zero power series solution of $(E_p) : x(y'' - y') + py = 0$, with coefficients satisfying $n(n+1)a_{n+1} = (n-p)a_n$ for all $n \in \mathbb{N}^*$. Show that there exists a natural integer $q > p$ such that, for all integer $n \geqslant q$, $$\left| a _ { n + 1 } \right| \geqslant \frac { \left| a _ { n } \right| } { 2 ( n + 1 ) }.$$

For $p \notin \mathbb{N}^*$, $p \neq 0$, let $f(x) = \sum_{n=0}^{+\infty} a_n x^n$ be a non-zero power series solution of $(E_p)$, with $q$ a natural integer such that $q > p$ and $\left| a _ { n } \right| \geqslant \frac { q ! \left| a _ { q } \right| } { 2 ^ { n - q } n ! }$ for all $n \geqslant q$. Show that the function $\psi : \left\lvert\, \begin{array} { c c c } \mathbb { R } _ { + } ^ { * } & \rightarrow & \mathbb { R } \\ x & \mapsto & \sum _ { n = q } ^ { + \infty } \left| a _ { n } \right| x ^ { n } \end{array} \right.$ is not an element of $E$.

For $p \notin \mathbb{N}^*$, $p \neq 0$, let $f(x) = \sum_{n=0}^{+\infty} a_n x^n$ be a non-zero power series solution of $(E_p)$. Using the result of Question 39, deduce finally that the function $f$ is not an element of $E$.

To each function $f \in E$, we associate the endomorphism $U$ of $E$. Let $\lambda \in \mathbb{R}^*$ be an eigenvalue of $U$ with eigenvector $f$, which is a solution of $(E_{1/\lambda})$. We assume that $f$ is developable as a power series on $\mathbb { R } _ { + } ^ { * }$, that is, there exists a power series $\sum _ { n \geqslant 0 } a _ { n } x ^ { n }$ of infinite radius of convergence such that $$\forall x \in \mathbb { R } _ { + } ^ { * } , \quad f ( x ) = \sum _ { n = 0 } ^ { + \infty } a _ { n } x ^ { n } .$$ Using the results of Part IV, show that the only possible eigenvalues of $U$ are of the form $\lambda = 1 / p$ with $p \in \mathbb { N } ^ { * }$.

Determine the domain of definition of $\sigma$ and justify that $\sigma$ is continuous on it, where $\sigma(x) = \sum_{k=1}^{+\infty} \frac{x^k}{k^2}$.

For all $n \in \mathbb{N}$ and all $z \in \mathbb{C}$, we define $$B_{n}(z) = n! \int_{0}^{1} \frac{\mathrm{e}^{z\omega(t)}}{(\mathrm{e}^{\omega(t)} - 1)\omega(t)^{n-1}} \,\mathrm{d}t$$ where $\omega(t) = e^{2i\pi t}$, and for all $p \in \mathbb{Z}$, $$I_{p} = \int_{0}^{1} \frac{\omega(t)^{p+1}}{\mathrm{e}^{\omega(t)} - 1} \,\mathrm{d}t.$$ Show that, for all $n \in \mathbb{N}$ and all $z \in \mathbb{C}$, $$B_{n}(z) = n! \sum_{k=0}^{n} \frac{z^{k}}{k!} I_{k-n}.$$ Deduce that $B_{n}$ is a monic polynomial of degree $n$.

Let $f(x) = \sum_{n=0}^{\infty} c_n x^n \in \mathbf{Q}\llbracket x \rrbracket$ be a power series whose coefficients $c_n$ are integers. Show that if there exists a real number $\alpha \geq 1$ such that the numerical series $\sum_{n=0}^{\infty} c_n \alpha^n$ converges, then $f$ is a polynomial.

Let $\sum _ { n \geqslant 0 } a _ { n } z ^ { n }$ be a power series with radius of convergence $R \geqslant 1$ and sum $f$. We denote $$\Delta _ { \theta _ { 0 } } = \left\{ z \in \mathbb { C } ; | z | < 1 \text { and } \exists \rho > 0 , \exists \theta \in \left[ - \theta _ { 0 } , \theta _ { 0 } \right] , z = 1 - \rho e ^ { i \theta } \right\}$$ for $\theta _ { 0 } \in [ 0 , \pi / 2 [$.
The purpose of this question is to prove that $$\left( \sum _ { n \geqslant 0 } a _ { n } \text { converges } \right) \Rightarrow \left( \lim _ { \substack { z \rightarrow 1 \\ z \in \Delta _ { \theta _ { 0 } } } } f ( z ) = \sum _ { n = 0 } ^ { + \infty } a _ { n } \right) \qquad \text{(Abel)}$$
(a) Prove (Abel) for $R > 1$.
From now on, we assume that $R = 1$ and that $\sum _ { n \geqslant 0 } a _ { n }$ converges, and we are given a $\theta _ { 0 } \in [ 0 , \pi / 2 [$.
(b) Prove that for all $N \in \mathbb { N } ^ { * }$ and $z \in \mathbb { C } , | z | < 1$, we have $$\sum _ { n = 0 } ^ { N } a _ { n } z ^ { n } - S _ { N } = ( z - 1 ) \sum _ { n = 0 } ^ { N - 1 } R _ { n } z ^ { n } - R _ { N } \left( z ^ { N } - 1 \right)$$
(c) Deduce that for all $z \in \mathbb { C } , | z | < 1$, we have $$f ( z ) - S = ( z - 1 ) \sum _ { n = 0 } ^ { + \infty } R _ { n } z ^ { n }$$
(d) Let $\varepsilon > 0$. Prove that there exists $N _ { 0 } \in \mathbb { N }$ such that for all $z \in \mathbb { C } , | z | < 1$ $$| f ( z ) - S | \leqslant | z - 1 | \sum _ { n = 0 } ^ { N _ { 0 } } \left| R _ { n } \right| + \varepsilon \frac { | z - 1 | } { 1 - | z | }$$
(e) Prove that there exists $\rho \left( \theta _ { 0 } \right) > 0$ such that for all $z \in \Delta _ { \theta _ { 0 } }$ of the form $z = 1 - \rho e ^ { i \theta }$ with $0 < \rho \leqslant \rho \left( \theta _ { 0 } \right)$, we have $$\frac { | z - 1 | } { 1 - | z | } \leqslant \frac { 2 } { \cos \left( \theta _ { 0 } \right) }$$ Deduce (Abel).

Exhibit a power series $\sum _ { n \geqslant 0 } a _ { n } z ^ { n }$ with radius of convergence 1 and sum $f$, such that $f ( z )$ converges when $z \rightarrow 1 , | z | < 1$ and such that the series $\sum _ { n \geqslant 0 } a _ { n }$ does not converge.

Let $\sum _ { n \geqslant 0 } a _ { n } z ^ { n }$ be a power series with radius of convergence 1 and sum $f$. Let $S \in \mathbb { C }$. The purpose of this question is to prove that $$\left( \lim _ { \substack { x \rightarrow 1 ^ { - } \\ x \in \mathbb { R } } } f ( x ) = S \text { and } a _ { n } = o \left( \frac { 1 } { n } \right) \right) \Rightarrow \left( \sum _ { n \geqslant 0 } a _ { n } \text { converges and } \sum _ { n = 0 } ^ { + \infty } a _ { n } = S \right) . \quad \text{(Weak Tauberian)}$$
In the rest of this question we suppose that $\lim _ { \substack { x \rightarrow 1 ^ { - } \\ x \in \mathbb { R } } } f ( x ) = S$ and that $a _ { n } = o \left( \frac { 1 } { n } \right)$.
(a) Prove that for all $n \in \mathbb { N } ^ { * }$ and $x \in \left] 0,1 \right[$, we have $$\left| S _ { n } - f ( x ) \right| \leqslant ( 1 - x ) \sum _ { k = 1 } ^ { n } k \left| a _ { k } \right| + \frac { \sup _ { k > n } \left( k \left| a _ { k } \right| \right) } { n ( 1 - x ) }$$
(b) Deduce (Weak Tauberian) by specifying $x = x _ { n } = 1 - 1 / n$ for $n \in \mathbb { N } ^ { * }$.

Let $k \in \mathbb{N}^*$ and $(a_1, \ldots, a_k) \in (\mathbb{N}^*)^k$ a $k$-tuple of strictly positive integers. When $k \geq 2$, we assume they are coprime as a set. We define a function $P : \mathbb{N} \rightarrow \mathbb{C}$ by setting for all $n \in \mathbb{N}$: $$P(n) = \operatorname{Card}\left\{(n_1, \ldots, n_k) \in \mathbb{N}^k : n_1 a_1 + \cdots + n_k a_k = n\right\},$$ then we define the power series $F(x) = \sum_{n=0}^{\infty} P(n) x^n$.
Show that the radius of convergence of $F$ is greater than or equal to 1.